Ewald’s Method Revisited: Rapidly Convergent Series
Representations of Certain Green’s Functions
Vassilis G. Papanicolaou1
Suggested Running Head: Ewald’s Method Revisited
Complete Mailing Address of “Contact” Author (for office purposes):
Vassilis G. Papanicolaou
Department of Mathematics and Statistics
Wichita State University
Wichita, KS 67260-0033
tel. (316) 978–3936
FAX (316) 978–3748
E-mail: [email protected]
1
Department of Mathematics and Statistics, Wichita State University, Wichita,
KS 67260-0033
1
ABSTRACT
We propose a justification of Ewald’s method of obtaining rapidly convergent series for the Green’s function of the 3-dimensional Helmholtz equation.
Our point of view enables us to extend the method to Green’s functions for
the Helmholtz equation in certain domains of Rd , with quite general boundary conditions.
Key words. Helmholtz equation, Green’s function, Ewald’s method.
2
1 Introduction
In 1921 P. P. Ewald published a sensational paper (see [1]). He presented
a method that was used to transform the slowly convergent series of the
Green’s function of the 3-d Helmholtz equation, with Floquet (e.g. periodic)
boundary conditions in the x and y directions, to a series that converges very
rapidly.
Ewald’s method gives spectacular computational results and hence it has
been used extensively by physicist, engineers, and numerical analysts (as an
example, see [2]). However, it is not transparent at all. The reader is usually
left with the impression that it works by coincidence. The method is based
on the formula
µ
¶
Z ∞
eiKR
1
K2
2 2
=√
exp −R z + 2 dz,
R
4z
π −∞
which somehow comes “out of the blue”.
In this note we propose a point of view that sheds some understanding
to the underlying structure of Ewald’s approach. This enables us to easily
extend his method to any dimension and a variety of boundary conditions.
2 A Rapidly Convergent Series for the Green’s Function
Let L be a self-adjoint operator on the space L2 (D), where D is a domain
in Rd , such that inf σ(L) = λ0 , where σ(L) is the spectrum of L. If λ is a
complex number with < {λ} < λ0 , then
Z ∞
−1
(L − λ) =
eλt e−tL dt.
0
In terms of integral kernels, the above equation can be written as
Z ∞
G(x, y; λ) =
eλt p(t, x, y)dt,
(1)
0
where G(x, y; λ) is the Green’s function and p(t, x, y) the heat kernel associated to L (thus, the Green’s function is the Laplace transform of the heat
kernel, viewed as a function of t). The integral in (1) converges if
< {λ} < λ0
and diverges if < {λ} > λ0 (if < {λ} = λ0 , convergence is possible). On the
other hand, G(x, y; λ) is analytic in λ, on C \ σ(L).
If L = −∆, acting on Rd (hence σ(L) = [0, ∞); L is the “free” ddimensional negative Laplacian operator), then its heat kernel is
3
p(t, x, y) =
Ã
1
(4πt)d/2
|x − y|2
exp −
4t
!
.
Let Γd (x, y; λ) be the corresponding Green’s function, i.e. the integral kernel
of (L − λ)−1 . Then (1) gives
Ã
!
Z ∞
1
|x − y|2
def
Γd (x, y; λ) =
exp λt −
dt = Γd (|x − y| ; λ)
d/2
4t
(4πt)
0
(2)
(Γd (x, x; λ) = ∞, if d ≥ 2). By deforming the contour of the above integral,
we can get Γd (x, y; λ), for < {λ} ≥ 0, but one can do even better, since
Γd (|x − y| ; λ) can be computed explicitly. If d = 1, it is easy to see (by
solving the associated ordinary differential equation for Γ1 ) that
√
e− −λ|x−y|
√
Γ1 (x, y; λ) = Γ1 (|x − y| ; λ) =
.
2 −λ
Furthermore, from (2) one observes that
d
1
Γd (R; λ) =
Γd−2 (R; λ).
dλ
4π
From this equation one can compute Γd (x, y; λ) for all odd dimensions d:
√
e− −λ|x−y|
,
Γ3 (x, y; λ) =
4π |x − y|
etc.
If d is even, Γd (x, y; λ) is not an elementary function. One can find the radial
solutions of the equation
−∆u = λu + δ(x),
x ∈ Rd ,
to conclude that (for all d)
Γd (x, y; λ) =
1
(2π)d/2
(−λ)(d/4)−(1/2) |x − y|1−(d/2) K(d/2)−1
³√
´
−λ |x − y| ,
where Kν (·) is the modified Bessel function of the 2nd kind, of order ν.
Finally, we notice that Γd (R; λ) decays exponentially, as R → ∞. This is
easy to establish if d is odd, whereas, if d is even, one can, for example, use
(2) to get the estimate
|Γd (R; λ)| < |Γd+1 (R; λ)| + |Γd−1 (R; λ)|
4
(alternatively, one can do asymptotic analysis, as |x − y| → ∞, to the integral in (2)).
Now consider the domain D in Rd
D = (0, b1 ) × · · · × (0, br ) × Rd−r
(r is some integer between 1 and d), and let Lα = −∆ act on D with αFloquet-Bloch boundary conditions. This means that there are numbers
α1 , ..., αr such that, for u to be in the domain of Lα , we must have
u(x1 , ..., bj , ..., xd ) = eiαj bj u(x1 , ..., 0, ..., xd ),
∇u(x1 , ..., bj , ..., xd ) = eiαj bj ∇u(x1 , ..., 0, ..., xd ),
(3)
for all j = 1, ..., r. Without loss of generality we assume that
−
π
π
< αj ≤ ,
bj
bj
j = 1, ..., r.
The method of images gives that the Green’s function of Lα is
X
G(x, y; λ) =
e−iα·(mb) Γd (x + mb, y; λ),
Zr
m∈
where
α = (α1 , ..., αr , 0, ..., 0)
and
mb = (m1 b1 , ..., mr br , 0, ..., 0)
are considered in Rd . Using (2) in the above equation, we get
Ã
!
Z ∞
2
X
|x
−
y
+
mb|
1
exp λt −
dt,
G(x, y; λ) =
e−iα·(mb)
d/2
4t
(4πt)
0
r
m∈Z
(4)
where, for λ < 0 (or, more generally, < {λ} < 0), the series converges
absolutely, provided x 6= y (x and y are in D).
We now show how to obtain a series representation of G(x, y; λ), x 6=
y, that converges very rapidly, for all λ ∈ C \ N , where N is a discrete
(countable) set. First we write (following Ewald’s approach)
G(x, y; λ) = G1 (x, y; λ) + G2 (x, y; λ),
where
G1 (x, y; λ) =
X
Zr
m∈
Z
E
e−iα·(mb)
0
1
(4πt)d/2
5
Ã
|x − y + mb|2
exp λt −
4t
!
dt,
(5)
G2 (x, y; λ) =
X
m∈
Zr
Z
−iα·(mb)
∞
e
E
1
(4πt)d/2
Ã
|x − y + mb|2
exp λt −
4t
!
dt,
(6)
and E is a positive number. Observe that G1 (x, y; λ) and G2 (x, y; λ) are the
integral kernels of the operators
Z E
Z ∞
−t(Lα −λ)
e
dt
and
e−t(Lα −λ) dt = (Lα − λ)−1 e−E(Lα −λ) ,
0
E
respectively.
The series for G1 (x, y; λ), given in (5), is already rapidly convergent,
for all λ ∈ C ¡(as long as x¢ 6= y), since its general term decays (in m) like
C |mb|−2 exp − |mb|2 /4E .
In the case r = d, G2 (x, y; λ), being the integral kernel of (Lα − λ)−1 e−E(Lα −λ) ,
has the eigenfunction expansion
2
2
2
e−[|α| +4π |m/b| +4πα·(m/b)−λ]E
1 X
G2 (x, y; λ) =
ei[2π(m/b)+α]·(x−y) ,
2
2
2
|D|
|α| + 4π |m/b| + 4πα · (m/b) − λ
(7)
m∈Zd
where |D| = b1 b2 · · · bd is the volume of D and
m/b = (m1 /b1 , ..., md /bd ).
Formula (7) is valid for all λ 6= |α|2 + 4π 2 |m/b|2 + 4πα · (m/b), i.e. all
λ∈
/ σ(Lα ), at which G2 (x, y; λ) has simple poles. The terms of the series
above decay (in m) like
¡
¢
C |m/b|−2 exp −4π 2 |m/b|2 E ,
for any fixed λ ∈ C \ σ(Lα ).
Now, let us discuss the case r < d. For the series in (6) we first assume
that < {λ} < 0. In this case we can interchange summation and integration
and get
!
Z ∞ λt−|xr −yr |2 /4t à X
2
e
r
r
G2 (x, y; λ) =
e−|x −y +mb| /4t−iα·(mb) dt,
d/2
(4πt)
E
m∈Zr
(8)
where, for x = (x1 , ..., xd ), we have introduced
xr = (x1 , ..., xr , 0, ..., 0)
and
xr = (0, ..., 0, xr+1 , ..., xd ).
Next (following the spirit of Ewald’s approach) we invoke a lemma.
6
Lemma 1. If s > 0 and z, γ ∈ C, then
√
X 2
π iγz−γ 2 /4s2 X −π2 n2 /s2 −πγn/s2 +2πizn
−s (z+n)2 −iγn
e
=
e
e
.
s
n∈Z
n∈Z
Proof. Poisson Summation Formula (see, e.g. [3]) applied to the function
2 +Bx
f (x) = e−Ax
,
A > 0,
B ∈ C.
(in fact, the formula is also a theta function identity).
¥
The lemma implies immediately
X
2
r
r
e−|x −y +mb| /4t−iα·(mb) =
Zr
m∈
(4πt)r/2 iα·(xr −yr )−|α|2 t X −[4π2 |m/b|2 +4πα·(m/b)]t+2πi(xr −yr )·(m/b)
e
e
,
b1 · · · br
r
m∈Z
where
m/b = (m1 /b1 , ..., mr /br , 0, ..., 0).
Thus (8) becomes
G2 (x, y; λ) =
X ei[2π(m/b)+α]·(xr −yr ) Z ∞
2
2
2
r
r 2
dt
e−[|α| +4π |m/b| +4πα·(m/b)−λ]t−|x −y | /4t
.
(d−r)/2
b
1 · · · br
(4πt)
E
r
m∈Z
(9)
So far we have assumed < {λ} < 0. However, the terms of the series
above decay (in m) like
¡
¢
C |m/b|−2 exp −4π 2 |m/b|2 E ,
for any fixed λ ∈ C. Thus, in the tail of the series, we can take λ to be any
complex number. There seems to be a problem though with the first few
terms of the series, where we may have < {λ} ≥ |α|2 +4π 2 |m/b|2 +4πα·(m/b)
(so that the corresponding integrals diverge). The following lemma shows
how to overcome this problem.
Lemma 2. Let a, c, E be positive constants, and n a positive integer.
For < {λ} > 0 we set
Z ∞
dt
2
f (λ) =
e−(a−λ)t−c /4t
.
(4πt)n/2
E
7
Then the only singularity of f (λ) on C is a branch point at λ = a.
Proof. Define
Z
E
f0 (λ) =
e−(a−λ)t−c
0
2 /4t
dt
(4πt)n/2
.
Notice that f0 (λ) is entire in λ and
f (λ) + f0 (λ) =
1
(2π)n/2
³ √
´
(a − λ)(n/4)−(1/2) c1−(n/2) K(n/2)−1 c a − λ ,
where Kν (·) is the modified Bessel function of the 2nd kind, of order ν.
Thus, the exact type of the branch point of f (λ) at λ = a is known.
¥
Thus, if r < d, the lemma implies that the only singularities of G2 (x, y; λ),
viewed as a function of λ, are branch points at λ = |α|2 + 4π 2 |m/b|2 + 4πα ·
(m/b). These are also the singularities of G(x, y; λ). Notice that, if d−r > 2,
then G(x, y; λ) stays finite at these values of λ.
3 The Main Result
We summarize the main result established in the previous section (see (5)
and (9)):
Theorem. Consider the domain D in Rd
D = (0, b1 ) × · · · × (0, br ) × Rd−r
(r is some integer between 1 and d), and let Lα = −∆ act on D with αFloquet-Bloch boundary conditions (see (3)). Then, for any E > 0, the
Green’s function G(x, y; λ) of Lα has the following rapidly convergent series
representation
G(x, y; λ) = G1 (x, y; λ) + G2 (x, y; λ),
Ã
!
Z E
2
X
1
|x
−
y
+
mb|
G1 (x, y; λ) =
e−iα·(mb)
exp λt −
dt,
d/2
4t
(4πt)
0
r
m∈Z
G2 (x, y; λ) =
X ei[2π(m/b)+α]·(xr −yr ) Z ∞
2
2
2
r
r 2
dt
e−[|α| +4π |m/b| +4πα·(m/b)−λ]t−|x −y | /4t
,
(d−r)/2
b
1 · · · br
(4πt)
E
r
m∈Z
where α = (α1 , ..., αr , 0, ..., 0), mb = (m1 b1 , ..., mr br , 0, ..., 0),
m/b = (m1 /b1 , ..., mr /br , 0, ..., 0), xr = (x1 , ..., xr , 0, ..., 0),
and xr = (0, ..., 0, xr+1 , ..., xd ).
8
Remarks. (a) If n = 1, then
Z ∞
Z ∞
1
2
2
2
−(a−λ)t−c2 /4t dt
√
f (λ) =
e
= √ √ e−(a−λ)s −c /4s ds.
π E
4πt
E
√
It follows (differentiate with respect to E) that
√
µ
¶
p
ec a−λ
c
erfc
f (λ) = √
E (a − λ) + √
4 a−
λ
2 E
√
µ
¶
−c a−λ
p
e
c
erfc
,
+ √
E (a − λ) − √
4 a−λ
2 E
where
¡ √ ¢
erfc(z) = 1−erf(z) = 1− 2/ π
Z
z
¡
2
exp −s
¢
¡ √ ¢
ds = 2/ π
0
Z
∞
¡
¢
exp −s2 ds
z
(a well known entire function).
Thus, if r = d − 1, then we have a more explicit representation of the
integrals in the series for G2 (x, y; λ), in terms of the special function erfc(z).
(b) Lemma 2 implies that the representation of G2 (x, y; λ), is valid for
all λ 6= |α|2 + 4π 2 |m/b|2 + 4πα · (m/b), even though σ(Lα ) = [λ0 , ∞), where
λ0 = |α|2 .
(c) If we want to balance the decays of the terms of the series for
G1 (x, y; λ) and for G2 (x, y; λ), a reasonable choice is
· 2
¸1/2
1
b1 + b22 + · · · + b2r
E=
.
−2
−2
4π b−2
1 + b2 + · · · + br
(d) The case of Dirichlet or Neumann boundary conditions can be treated
similarly, since the corresponding Green’s function can be written as a sum
(alternating sum in the Dirichlet case) of “free” Green’s functions (method
of images) so we again have a formula very similar to (4).
(e) Let us take d = 3. If D = (0, b1 ) × (0, b2 ) × (0, b3 ) and α = (0, 0, 0),
we have
√
1 X e− −λ|x−y+mb|
,
G(x, y; λ) =
4π
|x − y + mb|
3
Z
m∈
G(x, y; λ) = G1 (x, y; λ) + G2 (x, y; λ),
where
G1 (x, y; λ) =
XZ
m∈
Z3
E
0
Ã
1
(4πt)3/2
9
|x − y + mb|2
exp λt −
4t
!
dt,
and, by (7),
2
2
1 X e−(4π |m/b| −λ)E 2iπm·(x−y)
G2 (x, y; λ) =
e
.
2 |m/b|2 − λ
b1 b2 b3
4π
3
m∈Z
(f) Again, let d = 3. Observe that (by substituting t = s−2 )
µ
¶
Z E
Z ∞
c2 dt
2
2
2
exp λt −
= 2 √ e−(c /4)s +λ/s ds.
3/2
4t t
0
1/ E
Thus, by (a) above we get
Z
E
0
µ
c2
exp λt −
4t
¶
dt
t3/2
µ
¶
µ
¶¸
√ · √
√
√
√
c
c
π ic λ
−ic λ
erfc √ + i λE + e
erfc √ − i λE .
=
e
c
2 E
2 E
(10)
For d = 3, the formula for G1 (x, y; λ) becomes
Ã
!
Z E
2
X
1
|x
−
y
+
mb|
G1 (x, y; λ) =
e−iα·(mb)
exp λt −
dt,
3/2
4t
(4πt)
0
3
m∈Z
and therefore, by (10), the integrals in the series terms can be computed
explicitly, in terms of the function erfc(z).
(g) If D = (0, b1 )×(0, b2 )×R (hence d = 3 and r = 2), and α = (α1 , α2 , 0),
we obtain the case of the paper of Kirk Jordan et al. (see [2]). Notice that
this is the only case where the terms of both G1 (x, y; λ) and G2 (x, y; λ) can
be computed in terms of the function erfc(z).
(h) In the case d > 1 we have that G(x, y; λ) → ∞, as |x − y| → 0. All
the singular behavior of G(x, y; λ) is captured in the term of the series for
G1 (x, y; λ) that corresponds to m = 0.
APPENDIX
The “master” formula in Ewald’s work [1],
µ
¶
Z ∞
eiKR
1
K2
2 2
=√
exp −R z + 2 dz
R
4z
π −∞
(the formula is in the sense of analytic continuation with respect to K;
strictly speaking it is valid in the region <{K 2 } ≤ 0; alternatively we can
10
take K real in the formula above and think of the integral as a contour
integral where the contour is the entire x-axis except for a small interval
containing 0 which can be replaced by a small semicircle avoiding 0—see
also [2]), can be seen as a variant of (2) for d = 3. An alternative, more
direct way to justify this formula is by applying the corollary of the (wellknown) lemma, given below, to the function
2
f (x) = e−x .
Lemma 3. If f ∈ L1 (R), then
¶
Z ∞
Z ∞ µ
1
f x−
dx =
f (x) dx.
x
−∞
−∞
Proof. By using the substitution u = −1/x we obtain
¶
¶
Z ∞ µ
Z 0 µ
1
1 du
f x−
f u−
dx =
x
u u2
0
−∞
and
¶
¶
Z ∞ µ
1
1 du
f x−
f u−
dx =
.
x
u u2
−∞
0
Adding up and replacing the dummy variable u by x we get
µ
¶
¶
¶µ
¶
Z ∞ µ
Z
Z ∞ µ
1
1
1 dx
1 ∞
1
f x−
f x−
dx =
f x−
=
1 + 2 dx.
x
x x2
2 −∞
x
x
−∞
−∞
Z
0
µ
Now, the substitution u = x − 1/x gives
¶µ
¶
¶µ
¶
Z 0 µ
Z ∞
Z ∞ µ
1
1
1
1
f x−
f (u) du
f x−
1 + 2 dx =
1 + 2 dx =
x
x
x
x
−∞
−∞
0
and the proof is completed.
¥
Corollary. If f ∈ L1 (R), a > 0, and b ≥ 0, then
¶
Z ∞ µ
Z
1 ∞
b
dx =
f ax −
f (x) dx
x
a −∞
−∞
(thus the first integral does not depend on b).
Proof. The case b = 0 is obvious. If b > 0, the substitution u = (ab)1/2 x
gives
µ
¶
µ ¶1/2 Z ∞ ·
¶¸
Z ∞ µ
√
1
b
b
f
ab u −
dx =
du.
f ax −
x
a
u
−∞
−∞
11
Thus, by the previous lemma
¶
µ ¶1/2 Z ∞ ³
Z ∞ µ
√ ´
b
b
f ax −
dx =
f u ab du
x
a
−∞
−∞
and the proof is finished by substituting u = (ab)−1/2 x in the integral of the
right hand side.
¥
12
Acknowledgements. The author wants to thank Professors Mansoor
Haider and Stephanos Venakides for finding a serious misprint in Remark
(a) and Formula (10) of the original manuscript. We also want to thank
Professor Peter Kuchment for his encouragement and suggestions.
13
References
[1]P. P. Ewald, Die Berechnung Optischer und Electrostatischer Gitterpotentiale, Annalen der Physik, 64, 253–287 (1921); Translated by A. Cornell,
Atomics International Library, 1964.
[2]K. E. Jordan, G. R. Richter, and P. Sheng, An Efficient Numerical Evaluation of the Green’s Function for the Helmholtz Operator on Periodic
Structures, Journal of Computational Physics, 63, 222–235 (1986).
[3]Y. Katznelson, An Introduction to Harmonic Analysis, Dover Publications, Inc., New York, 1976.
14